
Izv. IMI UdGU, 2020, Volume 55, Pages 33–41
(Mi iimi389)




MATHEMATICS
Asymptotic expansion of a solution of a singularly perturbed optimal control problem with a convex integral quality index, whose terminal part additively depends on slow and fast variables
A. R. Danilin^{a}, A. A. Shaburov^{b} ^{a} Institute
of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16,
Yekaterinburg, 620108, Russia
^{b} Ural Federal University, ul. Mira, 19, Yekaterinburg, 620002, Russia
Abstract:
The paper deals with the problem of optimal control with a Boltz–type quality index over a finite time
interval for a linear steady–state control system in the class of piecewise continuous controls with smooth
control constraints. In particular, we study the problem of controlling the motion of a system of small
mass points under the action of a bounded force. The terminal part of the convex integral quality index
additively depends on slow and fast variables, and the integral term is a strictly convex function of control
variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary
and sufficient condition for optimality. The main difference between this study and previous works is that
the equation contains the zero matrix of fast variables and, thus, the results of A. B. Vasilieva on the
asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady–state
system satisfies the condition of complete controllability. The article shows that problems of optimal
control with a convex integral quality index are more regular than time–optimal problems.
Keywords:
optimal control, singularly perturbed problems, asymptotic expansion, small parameter.
DOI:
https://doi.org/10.35634/2226359420205503
Full text:
PDF file (136 kB)
References:
PDF file
HTML file
Bibliographic databases:
UDC:
517.977
MSC: 49N05, 93C70 Received: 01.03.2020
Citation:
A. R. Danilin, A. A. Shaburov, “Asymptotic expansion of a solution of a singularly perturbed optimal control problem with a convex integral quality index, whose terminal part additively depends on slow and fast variables”, Izv. IMI UdGU, 55 (2020), 33–41
Citation in format AMSBIB
\Bibitem{DanSha20}
\by A.~R.~Danilin, A.~A.~Shaburov
\paper Asymptotic expansion of a solution of a singularly perturbed optimal control problem with a convex integral quality index, whose terminal part additively depends on slow and fast variables
\jour Izv. IMI UdGU
\yr 2020
\vol 55
\pages 3341
\mathnet{http://mi.mathnet.ru/iimi389}
\crossref{https://doi.org/10.35634/2226359420205503}
Linking options:
http://mi.mathnet.ru/eng/iimi389 http://mi.mathnet.ru/eng/iimi/v55/p33
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  106  Full text:  32  References:  5 
